Hopf algebras in algebraic topology

I’m going to talk today about a little bit of algebra. There’s a beautiful relation to algebraic topology which I’ll touch on as well. (For a brief introduction to cup products, refer to section 7 of my notes here: http://www.mit.edu/~sanathd/papers/algebraic-topology.pdf.)

Definition: A Hopf algebra is a graded R-algebra A = \bigoplus_{n=0}A^n satisfying:

  1. There is an identity 1\in A^0 such that R\to A^0 given by r\mapsto r\cdot 1 is an isomorphism.
  2. There is a homomorphism of graded algebras \Delta:A\to A\otimes_R A, called the diagonal/coproduct, such that for \alpha\in A^n and n>0, we have: \displaystyle\Delta(\alpha) = \alpha\otimes 1 + 1\otimes \alpha + \sum_{i=1}^{n-1}\beta_i\otimes\gamma_{n-i}, where \beta_k,\gamma_k\in A^j.

For example: R[\alpha]. Because elements of R[\alpha] of dimension lower than \alpha are the elements of R, the terms \beta_k and \gamma_k are 0, and the diagonal is hence \Delta(\alpha) = \alpha\otimes 1 + 1\otimes \alpha.

Another example is the algebra k[\alpha]/(\alpha^n) where k is a field. In this case, \alpha is primitive, i.e., \Delta(\alpha) = \alpha\otimes 1 + 1\otimes\alpha. Therefore, we have:

\displaystyle\Delta(\alpha^n) = \alpha^n\otimes 1 + 1\otimes \alpha^n + \sum_{i=1}^{n-1}\binom{n}{i}\alpha^i\otimes\alpha^{n-i}

But because we’re thinking of the truncated polynomial ring, \alpha^n = 0, and hence:

\displaystyle\sum_{i=1}^{n-1}\binom{n}{i}\alpha^i\otimes\alpha^{n-i} = 0

Thus if 0<i<n, we must have \binom{n}{i} = 0. If k is of characteristic zero this is impossible. If k has nonzero characteristic p, then n must be a power of p (this can be proved using the fact that the freshman’s dream is true in characteristic p).

Yet another example is the exterior algebra \Lambda_R[\alpha] over R with an odd-dimensional generator. (This is the free R-module with basis generated by \alpha with multiplication defined by \alpha^2 = 0. For example, the cohomology ring \mathrm{H}^\ast(T^1;R)\cong\Lambda_R[\alpha].) This is a Hopf algebra with diagonal \Delta(\alpha) = \alpha\otimes 1 + 1\otimes\alpha; to show that it indeed forms a Hopf algebra we must only check that \Delta(\alpha^2)=0. But \Delta(\alpha^2) = \Delta(\alpha)^2, so consider (\alpha\otimes 1 + 1\otimes\alpha)^2. This is equal to \alpha^2\otimes 1 + 1\otimes\alpha^2 + (\alpha\otimes 1)(1\otimes\alpha). The latter term is 0 since \alpha has odd dimension, and \alpha^2 = 0 by definition, so we’re done.

One more rather important example: if A is a finitely generated free R-module in each dimension then A^\ast is a Hopf algebra, called the dual Hopf algebra (see Proposition 3C.10 in Hatcher). The product A^\ast\otimes A^\ast\to A^\ast is the dual of the diagonal \Delta:A\to A\otimes A, and the diagonal A^\ast\to A^\ast\otimes A^\ast is the dual of the product A\otimes A\to A.

Ok, so now onto topology. It turns out that there are natural examples of Hopf algebras in algebraic topology. We need two pieces of background. First, the notion of a H-space. Recall the notion of a topological group; this is a space X with a group structure such that the product X\times X\to X and the inverse map X\to X are continuous. Every Lie group is a topological group, for example. This motivates:

Definition: A H-space is a space with a continuous multiplication \mu:X\times X\to X and identity e\in X such that x\mapsto\mu(x,e) and x\mapsto \mu(e,x) are homotopic to the identity.

For example, every topological group is a H-space. Another classic and very important example is a loop space \Omega X, whose product is:

\displaystyle(\mu(\alpha,\beta))(t) = \begin{cases} \alpha(2t) & 0\leq t\leq 1/2\\ \beta(2t-1) & 1/2\leq t\leq 1 \end{cases}
where \alpha,\beta\in\Omega X

Note that this isn’t necessarily commutative. Now, suppose X is a H-space such that:

  1. \mathrm{H}^0(X;R)\cong R, which is implied, for example, if X is path-connected.
  2. \mathrm{H}^i(X;R) is finitely generated for each i, so the cross product \mathrm{H}^\ast(X;R)\otimes_R\mathrm{H}^\ast(X;R)\to \mathrm{H}^\ast(X\times X;R) is an isomorphism.

The multiplication \mu:X\times X\to X induces \mu^\ast:\mathrm{H}^\ast(X;R)\to \mathrm{H}^\ast(X\times X;R), giving a map:

\displaystyle \mathrm{H}^\ast(X;R)\xrightarrow{\Delta}\mathrm{H}^\ast(X;R)\otimes_R\mathrm{H}^\ast(X;R)

This is a map of R-algebras. Let i:X\to X\times X be the inclusion x\mapsto (e,x), and consider the commutative diagram:
hopf diagram

The map F satisfies F(\alpha\otimes 1)=\alpha and F(\alpha\otimes \beta) = 0 for |\beta|>0. By commutativity, \mu i \simeq 1, and hence F\Delta = 1. Hence the component of \Delta(\alpha) in the tensor H^0(X;R)\otimes H^n(X;R) is 1\otimes \alpha. Continuing like so, we get the following formula for \Delta(\alpha) where \beta_i,\gamma_i are of degree i:

\displaystyle \Delta(\alpha) = 1\otimes \alpha + \alpha\otimes 1 + \sum_{i=1}^{n-1}\beta_i\otimes \gamma_{n-i}

And we get the structure of a Hopf algebra! This is very cool. Let’s use the examples of Hopf algebras we computed early on to look at examples of H-spaces. So for finite n, we see that \mathbf{CP}^n is not a H-space by the example above about truncated polynomial rings. But, because polynomial rings are Hopf algebras, \mathbf{CP}^\infty is a H-space! Recall from homology:

Proposition: The boundary map in C_\ast(X\times Y) is given by d(e^i\times e^j) = de^i\times e_j + (-1)^ie^i\times de_j. The product of a boundary and cycle is a boundary because da\times b = d(a\times b) and a\times db = (-1)^id(a\times b) if da=0. This gives a bilinear map \mathrm{H}_i(X;R)\times \mathrm{H}_j(Y;R)\to \mathrm{H}_{i+j}(X\times Y;R), and hence a homomorphism \mathrm{H}_i(X;R)\otimes \mathrm{H}_j(Y;R)\to \mathrm{H}_{i+j}(X\times Y;R), which is the cross product in homology.

If X is a H-space, the composition \mathrm{H}_\ast(X;R)\otimes \mathrm{H}_\ast(X;R)\to \mathrm{H}_\ast(X\times X;R)\to \mathrm{H}_\ast(X;R) is called the Pontryagin product. This isn’t associative unless the product \mu:X\times X\to X is associative up to homotopy. We then have a natural example of dual Hopf algebras coming from algebraic topology (the result follows from the fact that the hypotheses imply that \mathrm{H}^n(X;R)\cong \mathrm{Hom}_R(\mathrm{H}_n(X;R),R).)!

Theorem: If X is a H-space whose multiplication is associative up to homotopy and \mathrm{H}^i(X;R) is a finitely generated free R-module for all i, then \mathrm{H}_\ast(X;R) with the Pontryagin product is the dual Hopf algebra of \mathrm{H}^\ast(X;R).

What about \mathbf{RP}^n; is it a H-space? If we choose k=\mathbf{Z}/2\mathbf{Z} in the example about truncated polynomial rings, we see that we must have n+1 a power of 2. The universal cover of a path-connected H-space is a H-space, and Adams showed that S^1,S^3,S^7 are the only spheres which are H-spaces. Since \mathbf{RP}^n=S^n/{\pm 1} for n=1,3,7 gets their H-space structures from S^1,S^3,S^7, and thus \mathbf{RP}^n can be a H-space only for n=1,3,7. A pretty cool result, huh? This depends on Adams’ theorem (the Hopf invariant one theorem) on the values of n for which S^n has a H-space structure, which was proved using K-theory! I plan on writing a post about this in the near future.

Best,
SKD

 

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